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Mathematics > Number Theory

Title:
Prime Number Races

Abstract: This is a survey article on prime number races. Chebyshev noticed in the
first half of the nineteenth century that for any given value of x, there
always seem to be more primes of the form 4n+3 less than x then there are of
the form 4n+1. Similar observations have been made with primes of the form 3n+2
and 3n+1, with primes of the form 10n+3/10n+7 and 10n+1/10n+9, and many others
besides. More generally, one can consider primes of the form qn+a, qn+b, qn+c,
>... for our favorite constants q, a, b, c, ... and try to figure out which
forms are "preferred" over the others. In this paper, we describe these
phenomena in greater detail and explain the efforts that have been made at
understanding them.